Abstract

From a problem in elasticity that uses a nonlinear stress-strain relation, we derive an equation featuring the one-dimensional p-Laplacian operator with a periodic potential term. This equation is nonlinear, but homogeneous, and we derive a modified Pruefer transform to convert this second-order equation into a two-dimensional first-order system, with radial and angular components. The homogeneity of the original equation is reflected in the complete independence of the angular equation on any radial terms. This allows us to restate conditions of periodic behaviour in terms of the angular component only.
Using these techniques, we compare the nonlinear equation with its linear counterpart, the equation featuring the standard Laplacian operator. This linear equation can also be converted into a first-order system, the linearity of which allows the effect of the equation acting over one period to be written as a constant-valued matrix. This gives a certain structure to the linear equation, which is almost completely absent from the nonlinear case.
The p-Laplace equation with a constant potential has solutions that behave analogously to the trigonometric functions. We detail methods of approximating these functions and their inverses, along with proving accuracy bounds. In turn, we use these to approximate an asymptotic average of the increase in the angular component as t approaches infinity. This function, called the rotation number, is dependent only on a spectral parameter in our equation, and gives information about the stability of the solutions of the equation at that spectral value.
In the linear case, the spectral values that give periodic and anti-periodic behaviour can be characterised exactly as the boundary points on intervals over which this function is constant. These values also separate values of the spectral parameter that give bounded and unbounded behaviour. We shall show that this characterisation is no longer true in the nonlinear case, specifically that periodic behaviour can stem from spectral values inside these intervals, and that the intervals can occur outside of the bounds of the (anti-)periodic spectral values.